Tiny black bugs on a mattress we just bought...please help?

ok we just bought a mattress from someone...it is in great condition, not one stain on it. we only have a car and so we had to strap the mattress to our car for about an hour ride home. when we got home we noticed very very tiny black bugs all over the bed frame and mattress. they are so tiny that when you just touch them they squish. my husband had one crawl on his hand and it jumped off so he was thinking it might be a flea because the people had dogs that we got the mattress from, but our cats had fleas before and these bugs are WAY smaller. i really dont think they are bed bugs or fleas, but what else can they be and how do we get rid of them? is it possible that they all came from being on top of the car on the ride home??